What Are the Differences between Smoke Collision Types?
This is my theory:

"Static" means a non-animated, non-moving object

"Rigid" means a non-animated, (possibly) moving object

"Animated" means an animated (shape keys, armatures, etc.), (possibly) moving object.

Tests won't resolve the issue, since, as the original question points out, they're currently all the same. 

Here's my theory: I've heard talk about using volumetrics in the game engine.  When that happens, the interactions between static, rigid, and animated objects is going to be essential - it would be a waste to calculate everything if you weren't having the smoke interact with animated objects, for instance.  My theory is that they are currently "the same" because it does not matter outside of the game engine - it doesn't have to be specified, and resources are not delegated real-time as they are in the game engine.  But in the game engine, you DO need to label an object as static, rigid, or animated - and that goes for smoke as well.  Essentially, the technical difference between all of the above would be the same for any other object in the game engine - a necessary setting that tells physics how to interact with it to optimize resources.